(_a_) Let it be laid down that the result is to depend on their _relative_ knowledge of each subject, so that, whether their marks, for French, be "1, 2" or "100, 200," the result will be the same: and let it also be laid down that, if they get equal marks on 2 papers, the final marks are to have the same ratio as those of the 3rd paper. This is a case of ordinary Double Rule of Three. We multiply _A_'s 3 marks together, and do the same for _B_. Note that, if _A_ gets a single "0," his final mark is "0," even if he gets full marks for 2 papers while _B_ gets only one mark for each paper. This of course would be very unfair on _A_, though a correct solution under the given conditions. (_b_) The result is to depend, as before, on _relative_ knowledge; but French is to have twice as much weight as German or Italian. This is an unusual form of question. I should be inclined to say "the resulting ratio is to be nearer to the French ratio than if we multiplied as in (_a_), and so much nearer that it would be necessary to use the other multipliers _twice_ to produce the same result as in (_a_):" _e.g._ if the French Ratio were 9/10, and the others 4/9, 1/9 so that the ultimate ratio, by method (_a_), would be 2/45, I should multiply instead by 2/3, 1/3, giving the result, 1/3 which is nearer to 9/10 than if he had used method (_a_). (_c_) The result is to depend on _actual_ amount of knowledge of the 3 subjects collectively. Here we have to ask two questions. (1) What is to be the "unit" (_i.e._ "standard to measure by") in each subject? (2) Are these units to be of equal, or unequal value? The usual "unit" is the knowledge shown by answering the whole paper correctly; calling this "100," all lower amounts are represented by numbers between "0" and "100." Then, if these units are to be of equal value, we simply add _A_'s 3 marks together, and do the same for _B_. (_d_) The conditions are the same as (_c_), but French is to have double weight. Here we simply double the French marks, and add as before. (_e_) French is to have such weight, that, if other marks be equal, the ultimate ratio is to be that of the French paper, so that a "0" in this would swamp the candidate: but the other two subjects are only to affect the result collectively, by the amount of knowledge shown, the two being reckoned of equal value. Here I should add _A_'s German and Italian marks together, and multiply by his French mark. But I need not go on: the problem may evidently be set with many varying conditions, each requiring its own method of solution. The Problem in Knot VI. was meant to belong to variety (_a_), and to make this clear, I inserted the following passage: "Usually the competitors differ in one point only. Thus, last year, Fifi and Gogo made the same number of scarves in the trial week, and they were equally light; but Fifi's were twice as warm as Gogo's, and she was pronounced twice as good."